The beta plane approximation, in oceanography and general fluid dynamics, is a simplified coordinate system for the equations of motion where the variation of the Coriolis parameter f with latitude is approximated by

f  = f₀ + βy

where f₀ is the value of f at the mid-latitude of the region and β the latitudinal gradient of f at that same latitude. This formalism is used to investigate both equatorial and mid-latitude phenomena (for which there are slightly different beta plane approximations) where f varies significantly over a few tens of degrees latitude. The beta plane approximation allows considerable simplification of the governing equations and therefore the use of analytical investigation methods. See Gill [1982].

The beta plane equations are obtained by introducing a background stratification into the shallow water equations, expanding them around a reference latitude θ₀ with respect to ε~ θ − θ₀, and keeping terms up to first order in ". This approximation introduces the horizonal coordinates

x = r₀ cos θ₀(φ − φ₀)

y = r₀(θ − θ₀)

and expands the Coriolis parameter as

f = f₀ + β₀y + • • •

where β₀ is the beta paramter at the reference latitude. The resulting equations (after Muller [1995])

are:

Image:Img62.png + Image:Img64.png - Image:Img66.png = Image:Img67.png

Image:Img68.png   +  Image:Img69.png   +  Image:Img70.png

0  =  Image:Img72.png

Image:Img73.png   =  Image:Img74.png

Image:Img75.png   +  Image:Img76.png

where (u, v,w) are the velocity components in the (x, y, z) directions, r₀ is the mean radius of the Earth, θ₀ is the reference latitude, f₀ = 2Ωsin θ₀ is the Coriolis parameter at the reference latitude, Image:Img81.png is the beta parameter at the reference latitude, ρ_* is a constant reference density, δp and δρ are motionally induced deviations from prescribed background fields, and N is the buoyancy frequency.

Further Reading

• J.R.Holton. 2004. An introduction to dynamical meteorology, Academic Press. ISBN 978-0123540157.
• Physical Oceanography Index